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Auction Theory II Recap First-Price Revenue Equivalence Optimal Auctions Auction Theory II Lecture 19 Auction Theory II Lecture 19, Slide 1 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 2 Recap First-Price Revenue Equivalence Optimal Auctions Motivation Auctions are any mechanisms for allocating resources among self-interested agents resource allocation is a fundamental problem in CS increasing importance of studying distributed systems with heterogeneous agents currency needn't be real money, just something scarce Auction Theory II Lecture 19, Slide 3 Recap First-Price Revenue Equivalence Optimal Auctions Intuitive comparison of 5 auctions Intuitive Comparison of 5 auctions English Dutch Japanese 1st-Price 2nd-Price Duration #bidders, starting #bidders, fixed fixed increment price, clock increment speed Info 2nd-highest winner’s all val’s but none none Revealed val; bounds bid winner’s on others Jump bids yes n/a no n/a n/a Price yes no yes no no Discovery Fill in “regret” after Regret no yes no yes no the fun game Auction• Theory How II should agents bid in these auctions? Lecture 19, Slide 4 Recap First-Price Revenue Equivalence Optimal Auctions Second-Price proof Theorem Truth-telling is a dominant strategy in a second-price auction. Proof. Assume that the other bidders bid in some arbitrary way. We must show that i's best response is always to bid truthfully. We'll break the proof into two cases: 1 Bidding honestly, i would win the auction 2 Bidding honestly, i would lose the auction Auction Theory II Lecture 19, Slide 5 Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions. Recap First-Price Revenue Equivalence Optimal Auctions English and Japanese auctions A much more complicated strategy space extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids intuitively, though, the revealed information doesn't make any difference in the IPV setting. Auction Theory II Lecture 19, Slide 6 Recap First-Price Revenue Equivalence Optimal Auctions English and Japanese auctions A much more complicated strategy space extensive form game bidders are able to condition their bids on information revealed by others in the case of English auctions, the ability to place jump bids intuitively, though, the revealed information doesn't make any difference in the IPV setting. Theorem Under the independent private values model (IPV), it is a dominant strategy for bidders to bid up to (and not beyond) their valuations in both Japanese and English auctions. Auction Theory II Lecture 19, Slide 6 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 7 Recap First-Price Revenue Equivalence Optimal Auctions First-Price and Dutch Theorem First-Price and Dutch auctions are strategically equivalent. In both first-price and Dutch, a bidder must decide on the amount he's willing to pay, conditional on having placed the highest bid. despite the fact that Dutch auctions are extensive-form games, the only thing a winning bidder knows about the others is that all of them have decided on lower bids e.g., he does not know what these bids are this is exactly the thing that a bidder in a first-price auction assumes when placing his bid anyway. Note that this is a stronger result than the connection between second-price and English. Auction Theory II Lecture 19, Slide 8 They should clearly bid less than their valuations. There's a tradeoff between: probability of winning amount paid upon winning Bidders don't have a dominant strategy any more. Recap First-Price Revenue Equivalence Optimal Auctions Discussion So, why are both auction types held in practice? First-price auctions can be held asynchronously Dutch auctions are fast, and require minimal communication: only one bit needs to be transmitted from the bidders to the auctioneer. How should bidders bid in these auctions? Auction Theory II Lecture 19, Slide 9 Recap First-Price Revenue Equivalence Optimal Auctions Discussion So, why are both auction types held in practice? First-price auctions can be held asynchronously Dutch auctions are fast, and require minimal communication: only one bit needs to be transmitted from the bidders to the auctioneer. How should bidders bid in these auctions? They should clearly bid less than their valuations. There's a tradeoff between: probability of winning amount paid upon winning Bidders don't have a dominant strategy any more. Auction Theory II Lecture 19, Slide 9 Recap First-Price Revenue Equivalence Optimal Auctions Analysis Theorem In a first-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0; 1], ( 2 v1; 2 v2) is a Bayes-Nash equilibrium strategy profile. Proof. 1 Assume that bidder 2 bids 2 v2, and bidder 1 bids s1. From the fact that v2 was drawn from a uniform distribution, all values of v2 between 0 and 1 are equally likely. Bidder 1's expected utility is Z 1 E[u1] = u1dv2: (1) 0 Note that the integral in Equation (1) can be broken up into two smaller integrals that differ on whether or not player 1 wins the auction. Z 2s1 Z 1 E[u1] = u1dv2 + u1dv2 0 2s1 Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions Analysis Theorem In a first-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0; 1], ( 2 v1; 2 v2) is a Bayes-Nash equilibrium strategy profile. Proof (continued). 1 We can now substitute in values for u1. In the first case, because 2 bids 2 v2, 1 wins when v2 < 2s1, and gains utility v1 − s1. In the second case 1 loses and gains utility 0. Observe that we can ignore the case where the agents have the same valuation, because this occurs with probability zero. Z 2s1 Z 1 E[u1] = (v1 − s1)dv2 + (0)dv2 0 2s1 2s1 = (v1 − s1)v2 0 2 = 2v1s1 − 2s1 (2) Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions Analysis Theorem In a first-price auction with two risk-neutral bidders whose valuations are drawn 1 1 independently and uniformly at random from [0; 1], ( 2 v1; 2 v2) is a Bayes-Nash equilibrium strategy profile. Proof (continued). We can find bidder 1's best response to bidder 2's strategy by taking the derivative of Equation (2) and setting it equal to zero: @ 2 (2v1s1 − 2s1) = 0 @s1 2v1 − 4s1 = 0 1 s = v 1 2 1 Thus when player 2 is bidding half her valuation, player 1's best strategy is to bid half his valuation. The calculation of the optimal bid for player 2 is analogous, given the symmetry of the game and the equilibrium. Auction Theory II Lecture 19, Slide 10 Recap First-Price Revenue Equivalence Optimal Auctions More than two bidders Very narrow result: two bidders, uniform valuations. Still, first-price auctions are not incentive compatible hence, unsurprisingly, not equivalent to second-price auctions Theorem In a first-price sealed bid auction with n risk-neutral agents whose valuations are independently drawn from a uniform distribution on the same bounded interval of the real numbers, the unique symmetric equilibrium is given by the strategy profile n−1 n−1 ( n v1;:::; n vn). proven using a similar argument, but more involved calculus a broader problem: that proof only showed how to verify an equilibrium strategy. How do we identify one in the first place? Auction Theory II Lecture 19, Slide 11 Recap First-Price Revenue Equivalence Optimal Auctions Lecture Overview 1 Recap 2 First-Price Auctions 3 Revenue Equivalence 4 Optimal Auctions Auction Theory II Lecture 19, Slide 12 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence Which auction should an auctioneer choose? To some extent, it doesn't matter... Theorem (Revenue Equivalence Theorem) Assume that each of n risk-neutral agents has an independent private valuation for a single good at auction, drawn from a common cumulative distribution F (v) that is strictly increasing and atomless on [v; v¯]. Then any auction mechanism in which the good will be allocated to the agent with the highest valuation; and any agent with valuation v has an expected utility of zero; yields the same expected revenue, and hence results in any bidder with valuation v making the same expected payment. Auction Theory II Lecture 19, Slide 13 Recap First-Price Revenue Equivalence Optimal Auctions Revenue Equivalence Proof Proof. Consider any mechanism (direct or indirect) for allocating the good. Let ui(vi) be i's expected utility given true valuation vi, assuming that all agents including i follow their equilibrium strategies. Let Pi(vi) be i's probability of being awarded the good given (a) that his true type is vi; (b) that he follows the equilibrium strategy for an agent with type vi; and (c) that all other agents follow their equilibrium strategies. ui(vi) = viPi(vi) − E[payment by type vi of player i] (1) From the definition of equilibrium, for any other valuation v^i that i could have, ui(vi) ≥ ui(^vi) + (vi − v^i)Pi(^vi): (2) To understand Equation (2), observe that if i followed the equilibrium strategy for a player with valuation v^i rather than for a player with his (true) valuation vi, i would make all the same payments and would win the good with the same probability as an agent with valuation v^i.
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